The so called Ramsey test is a semantic recipe for determining
whether a conditional proposition is acceptable in a given state of belief.
Informally, it can be formulated as follows:
(RT) Accept a proposition of the form "if A, then C" in a state of
belief K, if and only if the minimal change of K needed to
accept A also requires accepting C.
In Gärdenfors (1986) it was shown that the Ramsey test is, in the context of
some other weak conditions, on pain of triviality incompatible with the
following principle, which was there called the preservation criterion:
(P) If a proposition B is accepted in a given state of belief K and the
proposition A is consistent with the beliefs in K, then B is still
accepted in the minimal change of K needed to accept A.
(RT) provides a necessary and
sufficient criterion for when a 'positive' conditional should be included in a
belief state, but it does not say anything about when the negation of a
conditional sentence should be accepted. A very natural candidate for this
purpose is the following negative Ramsey test:
(NRT) Accept the negation of a proposition of the form "if A, then C"
in a consistent state of belief K, if and only if the minimal
change of K needed to accept A does not require accepting C.
This note shows that (NRT) leads to triviality
results even in the absence of additional conditions like (P).